Why do we have -1/g^2 * dg and not just -1/g^2? (This confused me at first). Why is the derivative negative? As dg increases, the denominator gets larger, the total value gets smaller, so we're actually shrinking (1/3 to 1/4 is a shrink of 1/12). The total change = dg * rate, or dg * (-1/g^2).g changes by dg, so 1/g becomes 1/(g + dg).The difference is negative, because the new value (1/4) is smaller than the original (1/3). (This is useful as a general fact: The change from 1/100 to 1/101 = one ten thousandth) If we make our derivative model perfect, and assume there's no difference between neighbors, the +1 goes away and we get: And the difference between "neighbors" (like 1/3 and 1/4) will be 1 / common denominator, aka 1 / (x * (x + 1)). ![]() How does this work? We get the common denominator: for 1/3 and 1/4, it's 1/12. What's the difference between 1/4 and 1/3? 1/12.We want the difference between neighboring values of 1/g: 1/g and 1/(g + dg). But what is dm (how much 1/g changed) in terms of dg (how much g changed)? m changes by dm, contributing area dm * f = ?.f changes by df, contributing area df * m = df * (1 / g).We just want to combine two perspectives: Inside function m is a division, but ignore that for a minute. So let's pretend 1/g is a separate function, m. but how does 1/g behave?Ĭhain rule to the rescue! We can wrap up 1/g into a nice, clean variable and then "zoom in" to see that yes, it has a division inside. Input x changes off on the side (by dx), so f and g change (by df and dg). We have a rectangle, we have area, but the sides are "f" and "1/g". The key is to see division as a type of multiplication: It's time to visualize the division rule (who says "quotient" in real life?). ![]() Oh, maybe you memorized it with a song like "Low dee high, high dee low.", but that's not understanding! Onward! Division (Quotient Rule)Īh, the quotient rule - the one nobody remembers. The chain rule is special: we can "zoom into" a single derivative and rewrite it in terms of another input (like converting "miles per hour" to "miles per minute" - we're converting the "time" input).Īnd with that recap, let's build our intuition for the advanced derivative rules. The derivative rules (addition rule, product rule) give us the "overall wiggle" in terms of the parts. Now, we can make a bigger machine from smaller ones (h = f + g, h = f * g, etc.). The derivative, dy/dx, is how much "output wiggle" we get when we wiggle the input: Functions are a machine with an input (x) and output (y) lever. Last time we tackled derivatives with a "machine" metaphor.
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